Problem: Solve for $x$ : $ 6|x + 10| - 4 = 5|x + 10| + 3 $
Explanation: Subtract $ {5|x + 10|} $ from both sides: $ \begin{eqnarray} 6|x + 10| - 4 &=& 5|x + 10| + 3 \\ \\ { - 5|x + 10|} && { - 5|x + 10|} \\ \\ 1|x + 10| - 4 &=& 3 \end{eqnarray} $ Add ${4}$ to both sides: $ \begin{eqnarray} 1|x + 10| - 4 &=& 3 \\ \\ { + 4} &=& { + 4} \\ \\ 1|x + 10| &=& 7 \end{eqnarray} $ Simplify: $ |x + 10| = 7$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 10 = -7 $ or $ x + 10 = 7 $ Solve for the solution where $x + 10$ is negative: $ x + 10 = -7 $ Subtract ${10}$ from both sides: $ \begin{eqnarray} x + 10 &=& -7 \\ \\ {- 10} && {- 10} \\ \\ x &=& -7 - 10 \end{eqnarray} $ $ x = -17 $ Then calculate the solution where $x + 10$ is positive: $ x + 10 = 7 $ Subtract ${10}$ from both sides: $ \begin{eqnarray} x + 10 &=& 7 \\ \\ {- 10} && {- 10} \\ \\ x &=& 7 - 10 \end{eqnarray} $ $ x = -3 $ Thus, the correct answer is $x = -17 $ or $x = -3 $.